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Completely regular semigroups with generalized strong semilattice decompositions. (English) Zbl 1103.20060

A regular semigroup \(S\) is called ‘cryptic’ if Green’s relation \(\mathcal H\) on \(S\) is a congruence. Clifford has shown that a regular semigroup is completely regular if and only if \(S\) is a semilattice \(Y\) of completely simple semigroups \(S_\alpha\) (\(\alpha\in Y\)) [see A. H. Clifford, Ann. Math. (2) 42, 1037-1049 (1941)]. Petrich and Reilly proved that for a cryptic semigroup \(S\), such that \(S/\mathcal H\) is a normal band, \(S\) can be expressed as a strong semilattice of completely simple semigroups [see M. Petrich and N. R. Reilly, Completely regular semigroups. Wiley (1999; Zbl 0967.20034)]. The authors introduce the concept of \(\rho G\)-strong semilattice of semigroups \(S_\alpha\), denoted by \(S=\rho G[Y;S_\alpha,\Phi_{\alpha,\beta}]\), with an equivalence relation \(\rho\) on \(S\) and a set of families of homomorphisms \(\Phi_{\alpha,\beta}\) (\(\alpha,\beta\in Y\), \(\alpha\geq\beta\)).
A completely regular semigroup \(S\) is a regular cryptogroup if and only if \(S\) is an \({\mathcal H}G\)-strong semilattice of completely simple semigroups. A completely regular semigroup \(S=[Y;S_\alpha]\) is a right quasi-normal cryptogroup if and only if \(S\) is an \({\mathcal L}G\)-strong semilattice of completely simple semigroups, i.e., \(S=[Y;S_\alpha,\Phi_{\alpha,\beta}]\). A completely regular semigroup \(S=[Y;S_\alpha]\) is a normal cryptogroup if and only if \(S\) is a \({\mathcal D}G\)-strong semilattice of completely simple semigroups, i.e., \(S=[Y;S_\alpha,\Phi_{\alpha,\beta}]\), where \(\mathcal D\) is the usual Green relation on \(S\).

MSC:

20M17 Regular semigroups
20M10 General structure theory for semigroups

Citations:

Zbl 0967.20034
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References:

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